US 7447614 B2
A method for modeling material behavior includes using empirical three dimensional non-uniform stress and strain data to train a self-organizing computational model such as a neural network. A laboratory device for measuring non-uniform stress and strain data from material includes an enclosure with an inclusion in it. As the enclosure is compressed, the inclusion induces a non-uniform state of stress and strain. A field testing device includes a body having a moveable section. When the body is inserted in a material and the moveable section moved, a non-uniform state of stress and strain can be characterized.
1. A method for modeling material behavior comprising the steps of:
inducing a non-uniform state of stress and strain in a material sample using a testing device that is operable to penetrate into the material and includes a body having at least two sections that act directly on the material, one of said at least two sections movable relative to the other section, the non-uniform state of stress and strain induced by inserting both of said testing device sections into the material to a first position and then moving said one section relative to said other section which remains substantially stationary at said first position;
measuring sample data with said testing device, said data being recordable; and,
training a self-organizing computational model with said recordable data to learn a non-uniform behavior for the material.
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moving both of said at least two sections to a first location and obtaining a first set of data; and,
moving said one of said at least two sections to an extended position of a predetermined distance D while said other section remains substantially stationary and obtaining a second data set.
18. A method for modeling material behavior as defined by
19. A method for modeling material behavior as defined by
20. A method for modeling material behavior comprising the steps of:
obtaining recordable empirical stress and strain data for said material over a period of time while said material is in a non-uniform state of stress and strain by measuring said material over said period of time to determine said empirical stress and strain data, and wherein said empirical stress and strain data represents said material non-uniform stress and strain; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for the material, said self-organizing computational model including a finite element analysis.
21. A method for modeling material behavior as defined by
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39. A method for modeling material behavior comprising the steps of:
obtaining empirical stress and strain data that is recordable in a memory and that measures displacement of said material over a period of time while said material is in a non-uniform state of stress and strain due to a wall abutting the material; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for said material.
40. A method for modeling material behavior comprising the steps of:
obtaining empirical stress and strain data for said material over a period of time that comprises at least a day while said material is in a non-uniform state of stress and strain, said empirical stress and strain data being recordable in a memory; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for said material, said self-organizing computational model including a finite element analysis.
41. A method for modeling material behavior as defined by
42. A method for modeling soil behavior comprising the steps of:
obtaining recordable empirical stress and strain displacement data for said soil over a period of time while said soil is in a non-uniform state of stress and strain as a structure is built on said soil; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for said soil.
43. A method for modeling soil behavior as defined by
44. A method for modeling soil behavior as defined by
45. A method for modeling material behavior comprising the steps of:
obtaining recordable empirical stress and strain data for said material over a period of time while said material is in a non-uniform state of stress and strain; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for said material, said self-organizing computational model comprising a nested adaptive neural network.
46. A method for modeling material behavior comprising the steps of:
inducing a non-uniform state of stress and strain in a material sample using a testing device to displace a first portion of said material sample and taking a first data measurement, and using only a portion of said testing device to displace a second portion of said material sample that is smaller than said first portion and taking a second data measurement;
said first data measurement and said second data measurement being recordable data; and,
training a self-organizing computational model with said data to learn a non-uniform behavior for the material sample.
47. A method for modeling material behavior as defined by
48. A method for modeling material behavior as defined by
49. A method for modeling material behavior as defined by
50. A method for modeling material behavior comprising the steps of:
inducing a non-uniform state of stress and strain in a material sample using a testing device, said testing device having a plurality of walls defining an enclosure for containing the sample, at least one of said walls moveable into said enclosure, said at least one wall configured to exert a force on the material sample contained in said enclosure as it moves into said enclosure and having one or more sensors for measuring said force;
measuring sample data with said testing device, said data being recordable in a memory; and,
training a self-organizing computational model with said recordable data to learn a non-uniform behavior for the material.
This application claims priority of U.S. Provisional Application Ser. No. 60/371,095, filed Apr. 9, 2002 under 35 U.S.C. §119.
This invention was made with Government assistance under National Science Foundation Grant No. MSS 92-14910, National Science Foundation Grant No. CMS 95-08462 and National Science Foundation Grant No. CMS 00-84556. The government has certain rights in the invention.
The present invention is related to methods and systems for modeling material behavior, including methods and systems for modeling stress and strain effects in soil.
In current engineering practice, the design and construction of structures such as tunnels, underground sewers, underground parking garages, foundations, buildings, and the like often begin with a characterization of the engineering response of soil due to the proposed construction. A numerical model may be used to predict static and dynamic soil structure reaction to construction activities. Generally, the characterization may include performing laboratory or field testing of the soil, and using the resultant data to develop a numerical model of the soil behavior. The numerical model may then be used to predict soil response to construction activities.
Most soil numerical models include constitutive models to represent soil behavior at the point or “element” level and describe soil stress-strain-strength response. Most constitutive models used in geomechanics are based on the concepts of plasticity. Early forms of a yield criterion include the Tresca and von Mises yield criteria used to represent material failure. These models are useful to establish a general framework for understanding material behavior but do not capture the complex behavior of geologic materials including non-linearity, hardening and softening, anisotropy, and strain rate dependence.
These constitutive models define the material behavior using a fixed set of equations and geometric objects that define the yield surface. An advantage of these types of constitutive relations relates to model generalization. Using a limited set of laboratory tests, a set of hypotheses about the general stress strain behavior of soil, and conservation laws of mechanics, the model clearly defines stress strain behavior even in regions of stress and strain space not covered by data from laboratory tests. A disadvantage associated with these models is there relative inflexibility. A given constitutive model has very limited learning capacity through adjustment of model properties. The fundamental shape of the stress-strain relationship does not change.
The constitutive numerical model may be calibrated using laboratory and/or field test data may be used to at least partially define the boundary values. The data may represent boundary value data. A variety of computer codes are available for providing a model through a solution to the boundary value problem. For example, a finite element or a finite difference method can be used to solve equilibrium equations that govern the boundary value problem. Also, a reference library of material models may be used to simulate the constitutive behavior of the soil. This general process as currently known, however, leaves many problems unresolved.
The data obtained from current laboratory and/or field-testing procedures and apparatuses, for example, are often not sufficient as a basis for comprehensive modeling. Field-testing may be performed using devices that penetrate the soil and measure axisymmetric states of stress. These include the Standard Penetration Test (SPT) or the Cone Penetrometer Test (CPT). These devices are generally unable to obtain some data important for soil modeling, such as the soil friction angle. The strain path method is one example of a computational method that has been used before to interpret soil properties around a CPT, an SPT and other in field-testing devices. Although this technique can be used to interpret strains, it is unable to develop the stress-strain relationship unless data such as the soil friction angle is assumed.
Additional problems are presented in the extent of testing required using presently known apparatuses and methods. Soil behavior is highly anisotropic and is difficult to fully characterize using a single test device. Historically, a large number of laboratory tests have been required using a range of test devices to obtain even a limited characterization of the soil properties. Known test devices generally measure only uniform states of stress and strain, and include a triaxial test apparatus, a plane strain apparatus, a hollow cylinder apparatus, a direct shear apparatus, a true triaxial apparatus, and similar devices. It is not uncommon for tests using the devices to number over one hundred in order to obtain a comprehensive data set.
It is also known to utilize observation of soil behavior under actual construction conditions may to verify soil models that have been developed using data from field and/or laboratory tests. If the behavior predicted by the model appears to be inconsistent with field observations, the model may be “tweaked” or otherwise adjusted to better correlate to the observed results. In the prior art, however, these model adjustments have only been made on an ad-hoc basis, with the result that they may be inconsistent and not accurately applied. Also, many models developed using laboratory and/or field-testing do not have the flexibility to be readily adjusted.
These and other problems remain unresolved in the art.
The present invention is directed to methods and program products for modeling material behavior, as well as testing devices for testing material. One embodiment of the invention is directed to a method that includes steps of inducing a non-uniform state of stress and strain in a material sample using a testing device, and measuring sample data with the testing device. A subsequent method step includes training a self-organizing computational model with the data to learn a non-uniform behavior for the material. The empirical data may be obtained, for example, from a field-testing device or a laboratory-testing device. The present invention is also directed to devices for measuring empirical strain and stress data. An additional exemplary invention embodiment is directed to a field-testing device, and another to a laboratory-testing device.
Still an additional invention embodiment is directed to a method for modeling material behavior, and includes the steps of obtaining empirical stress strain data for the material over a period of time while the material is in a non-uniform state of stress and strain, and training a self-organizing computational model with the empirical data with the data to learn a non-uniform behavior for the material. These and other aspects of the present invention will be discussed in the Detailed Description and Drawings that follow.
The present invention is directed to systems and methods for modeling material behavior. A preferred embodiment of the invention is directed to method for modeling material behavior that includes using a testing device to induce a non-uniform state of stress and strain in a material sample and to measure data with the testing device. The data is then used to train a self-organizing computational model to learn non-uniform stress and strain behavior of the material. Other aspects of the present invention are directed to devices for obtaining the empirical stress and strain data. The devices include field testing devices and methods, laboratory testing devices and methods, and field observation methods.
It will be appreciated that methods and devices of the invention lend themselves well to practice in the form of computer program products. For example, a method or device of the invention may include a computer program product that causes a processor-based device to carry out steps of a method of the invention or to operate as a device of the invention. Accordingly, discussion herein of systems, devices, and methods of the invention will be understood to also describe program products where appropriate.
A. Field Testing Devices
One aspect of the present invention is directed to a field testing device for obtaining stress and strain data from field material samples, such as soil.
A section 18 of the penotrometer 10 is movable, as best illustrated by
The drive 50 and the mechanical linkage 52 may include combinations of gears, chains, hydraulic fluids, chambers and pistons, and the like that are operable to cause the section 18 to move. A power supply (not shown) for powering the drive 50 may be an AC, DC, universal, or the like. Preferably the drive 50 is operable to move the moveable section 18 a known distance D.
The schematic diagrams of
Through these and other operations, the penotrometer 10 of the invention is useful to collect a variety of valuable stress and strain data. While the penotrometer 10 is moving through the material sample 72, some of the sensors 16 arranged along the cylindrical sidewall portion of the body 12 may be used to measure sleeve friction data. Sensors 16 arranged along the cone shaped penetrating end 14 may be used to measure tip resistance data. Some of the sensors 16 may also be used to measure pore water pressure.
Importantly, the penotrometer 10 of the present invention is also useful to collect data that can be used to characterize the material's response to non-uniform states of stress and strain. This is accomplished partly through movement of the penotrometer moving section 18. When the penotrometer 10 is in a stationary position and the moving section 18 in a retracted position, as illustrated by
Through these movements, a non-uniform and three-dimensional state of stress and displacement may be induced. Through knowledge of the distance D and the geometry of the moving section 18 and the penotrometer 10, these two sets of data may be interpreted to develop a model of the non-uniform stress and strain behavior of the material sample 72 through a method of the invention.
Those knowledgeable in the art will appreciate that the penotrometer 10 of
B. Laboratory Testing Devices
An additional aspect of the invention is directed to laboratory devices useful for measuring stress, strain, force, displacement, and like data for a material sample.
A controller such as a computer 112 is connected to the testing device 100 for controlling movement of the end wall 106, for controlling the sensors 110, for recording data from the sensors 110, and the like. A drive (not shown) may be provided for driving movement of the end wall 106. When the end wall 106 moves inward into the enclosure 104, compression of the material sample contained therein occurs. The end wall 104 is moved a predetermined distance D′ into the enclosure 104. This compressionary movement exerts a force on the boundary of the material and causes its displacement. At least some of the sensors 110 measure force at the material boundary while others measure displacement. Resultant data are recorded by the controller 112.
Importantly, the enclosure 104 includes an inclusion defined by the inverted dome 108 on the end wall 106. As used herein the term “inclusion” is intended to broadly refer to a formation or alteration that changes the volume of the enclosure 104. By way of example, the inverted dome 108 changes the otherwise smooth face of the endwall 106 and changes the volume of the enclosure 104. In operation, as the endwall 106 compresses the enclosure 104, the inverted dome 108 induces a non-uniform and three dimensional state of stress and strain in the material sample contained in the enclosure 104.
The sensors 110 may be used to collect displacement, force and like data useful to characterize the non-uniform three dimensional state of stress. In particular, one or more of the sensors 110 may be used to measure data with the end wall 106 in a first position, and then subsequently used to measure data at a second time when the end wall 106 is in a second position. One or more of the sensors 110 may also be used to dynamically measure data during movement of the end wall 106. Through knowledge of the distance D′, the geometry of the enclosure 104 and of the inverted dome 108, and measured boundary forces and displacements, a model of a non-uniform stress and strain for the material sample can be developed through a modeling method of the invention.
Those knowledgeable in the art will appreciate that the laboratory device 100 is only one preferred embodiment of the invention, and that many variations may be made within the scope of the invention. For example, the device of the invention is not limited to a general cylinder shape. Also, the inverted dome 108 of the device 100 could take many alternative forms. Other three dimensional geometric shapes such as pyramids, cubes, other polygons, and the like may be used. It is also contemplated that as an alternative to being a generally convex shape that extends outward form the end wall 106 into the material sample, other useful inclusions may include concave shapes that expand the volume of the enclosure 104.
By way of additional example,
Like the device 100 of
The presence of the inclusion 132 in the body of the material sample in the enclosure 124 induces a non-uniform state of stress and strain in the sample in generally the same manner as the inclusion 108 of
It will be appreciated that although the inclusion 132 has been illustrated as a three-dimensional disk, other free-floating inclusion shapes will be useful in the practice of the invention. Other shapes may be desirable depending on particular experiment purposes. For instance, it may be desirable to expose the same material sample to testing using differently sized/shaped inclusions. By way of example and not limitation, a substantially flat inclusion may be used, such as a sheet of metal or the like. Other three dimensional inclusions may also be used, with examples including but not limited to cubes, rectangles, spheres, pyramids, and the like. Irregular shaped inclusions are also contemplated.
It may also be useful to know the position of the free floating inclusion 132 in the material sample. This may be accomplished through use of a guide structure that may be useful to locate the inclusion in a known first position. An exemplary guide structure may include a wire or other thin support attached to a wall of the body 122 that can be used to place the inclusion at a precisely known position in the inclusion.
C. Self-organizing Computational Modeling Method
In addition to field and laboratory testing devices and methods, the present invention is also directed to a method for extracting a non-uniform material stress strain behavior from empirical data through use of a self-organizing computational model. As used herein, the term “self-organizing computational model” is intended to broadly refer to self-learning algorithms that can iteratively change to “learn” a solution to a problem. Several self-organizing computational models will be suitable for practice of the invention, with examples including evolutionary computational models, genetic algorithms, neural networks, and the like. The preferred self-organizing model is a neural network.
Neural networks are massively parallel computational models for knowledge representation and processing. They are capable of learning complex non-linear relationships and associations from a large body of data, and they have many desirable characteristics for engineering applications: they are robust, they are noise and fault-tolerant, they are self-organizing and adaptive, and they are capable of generalizing.
A simple neural network is schematically illustrated in
Each neuron in a given layer is connected to every neuron in the next layer, and each such connection has a connection weight associated with it; the knowledge stored in the neural network is represented by the set of connection weights. The neural network may be trained by presenting it with a comprehensive set of training cases. For example, a set of strain variables and the set of corresponding correct stress variables may be used. The connection weights are appropriately modified until the predicted stress variables agree satisfactorily with the correct stress variables.
A preferred method of the invention includes the step of training the neural network based model using empirical three dimensional stress and strain data that is obtained using a testing device that has induced a non-uniform state of stress and strain. The testing device may for example be a laboratory or field testing device of the invention. Accordingly, the empirical data is not used “directly” in modeling the material behavior, but is instead used as training data to teach a neural network based model. If the training data contains enough relevant information, the trained neural network can generalize material behavior to new loading cases.
Those skilled in the art will appreciate that training refers to the process of supervised learning during which training cases are presented to the neural network, and the connection weights are modified until correct results are obtained. Training is the adaptation process whereby neural networks adjust their connection weights so that they can produce the specified output in response to the specified input. This process may be thought of as a function mapping, that is, training is a procedure to develop a function, embedded in the connection weights, that can describe the relationship between the specified input and output. One advantage of neural networks function mapping is that it continuously adapts to new input and output. Once the neural network is trained, it can be used just as a conventional function. Use of a testing device of the invention to obtain training data ensures a comprehensive data set. The testing devices of the invention are useful to generate complex spatial patterns of stress and strain that are therefore rich in information.
Neural network training procedure can be thought of as equivalent to determining a conventional function that involves three major steps: a) observing the given data, b) choosing approximate function, and c) calibrating parameters of the function in conventional mapping function creation. The training data should implicitly contain the essential knowledge that describes the material behavior if the neural network is to be able to predict that full range of behavior. That is, the set of training cases should constitute a comprehensive set.
Those skilled in the art will appreciate that there are numerous appropriate methods for training a self-organizing computational model such as a neural network. In the preferred method of the invention, an autoprogressive training methodology is used. The autoprogressive training procedure is discussed in detail in “Autoprogressive Training of Neural Network Constitutive Models,” by J. Ghaboussi, D. Pecknold, M. Zhang, and R. Haj-Ali, Int. J. Num. Meth. Eng. 42, p 105-126 (1998). The autoprogressive training methodology provides an advantageous process for training a neural network to “learn” a global solution to a boundary value problem using only “local” boundary values. Thus neural network training exercises that might otherwise require training data from a large number of tests representing global values may be carried out with only one or a few tests using the autoprogressive training procedure.
Autoprogressive training is a process in which the neural network is itself an integral part of the iterative algorithm that is used to create the stress strain training cases from the empirical data. These stress strain training cases evolve, or progress, during this process: initially, they do not accurately represent the true material behavior. As the iterative process continues, the quality of the training cases improves. An important part of the training protocol therefore involves appropriately pruning the training set to reflect this evolution.
In order to illustrate the autoprogressive training algorithm, a hypothetical example is considered in which a one dimensional neural network stress strain model is developed using a structural response data for a simple truss having a plurality of individual members. The global response of the truss structure may be summarized as the horizontal displacement U at one side of the truss as a load P is applied to the other side. From this data, it is desired to train a neural network to learn the stress strain behavior of the individual members of the truss.
For purposes of this explanatory example, a finite element simulation is used to create hypothetical structural response data shown in
For this example, a simple neural network representation that uses the truss element current axial strain ε as input, and the current axial stress σ as output, as illustrated in
Where NN denotes the output of a multi-layer feed forward neural network, and the notation indicates that the stress σ is the output of the neural network. The first argument field specifies the input to the neural network, including loading history information if appropriate; the second argument describes the neural network architecture, i.e. the number of nodes in the input layer, the hidden layers, and the output layer, respectively, and its training history. Thus the specific neural network material model of
This indicates that the neural network has four layers: an input layer with one node; two hidden layers; and an output layer with one node. The two hidden layers each have two nodes at the start of the training process, and each has six nodes after training is completed
One preparatory step of training of the neural network material model is generally carried out in order to get the autoprogressive algorithm started. Rather than assigning random initial connection weights, it has been found convenient to initialize the neural network material model by pre-training it on a data set generated from a linear elastic constitutive model; this data set is retained in the training data generated during the incremental finite element simulation of the structural test.
In the case of the 1-D truss example, a plurality of pre-training cases was generated randomly to form the pre-training database. The neural network shown in
In order to create the stress strain training data for the neural network material model, having knowledge only of the global load vs. deflection response of the structure, a Finite Element (FE) model of the structural test specimen may be constructed. This model is used to carry out an incremental non-linear analysis of the structural test, using a neural network to represent the material stress strain behavior. In this mode, termed forward analysis, no training takes place; the neural network is simply used in the same manner that a conventional analytical material model is used.
Clearly, if the neural network has been only partially trained, or not trained at all, such a forward analysis will not provide accurate structural level responses. The autoprogressive algorithm provides a method for using the error in the predicted structural level response to generate approximate stress strain information with which to carry out a cycle of training; a new iteration of forward analysis with the updated neural network material model then yields a new set of approximate, but presumably improved, stress strain training cases. This iterative process is continued at each discrete load level until the structural level response is predicted accurately enough by forward analysis.
The two phases of the autoprogressive algorithm are the forward analysis phase and the training phase. The forward analysis phase that is carried out at each measured load level Pn. It consists of two main steps, each of which is a non-linear FE analysis.
Step 1: Load Controlled Analysis. At the nth load increment, the measured load increment ΔP=Pn−Pn−1 and the displacement increment ΔU is computed using:
in the current neural network model; where K is a tangent stiffness and the internal resisting force vector is computer from the stress at the end of the previous increment, which is in turn determined from the neural network by:
In this example, history points σn−2, εn−2 . . . are not included in the chosen neural network representation. The behavior of the structure is in general non-linear, and the displacement increment ΔU may be solved for as a standard non-linear problem. If a conjugate gradient algorithm is used, the tangent stiffness Kt need not be explicitly computed; alternatively, the tangent stiffness can be formed by computing element tangent stiffnesses via the neural network material model. This is done by computing the element tangent constitutive matrix, one column at a time, by using strain probes.
The iterative solution of Eqtn 1 is carried out in the usual way; for the jth iteration in the nth load increment, the following system of equations is formed:
Eqtn. 2 is solved to determine the iterative correction δUn j to the displacement increment, where εn j−1 is the current total strain, updated after iteration (j−1). Strain increments are computed at element integration points, the total strains are updated, and the stresses are updated by a forward pass through the neural network material model:
Iterations continue until specified convergence tolerances on, for example, the norm of the residual load (the right-hand side of Eqtn. 2) are met. When such convergence criteria are met, the stresses, strains and displacements are updated. The total stresses at this stage are denoted by σn−1+Δσn, the total strains by εn−1+Δεn, and the total nodal displacements by Un−1+ΔUn.
The relevant subset of the computed displacement vector Un, denoted by (Un vector), does not in general agree very closely with the measured displacements, denoted by (Un measured). The discrepancy, or error, is taken as:
Step 2: Displacement Controlled Analysis. The displacement error, Eqtn 3, is applied as an imposed structural boundary condition. This is considered as an additional incremental step from the current configuration. Iterations are carried out as usual until specified convergence tolerances are met, as in Step 1; the final strain increments δεn at the integration points are then computed and stored.
Collection of Training Cases: New stress strain sets, consisting of stresses determined from Step 1, and corresponding strains, computed from Step 1 and corrected with strain increments from Step 2 are collected. That is, the stresses are σn−1+Δσn and the corresponding strains are εn−1+Δεn+δεn.
Training phase of the autoprogressive algorithm: Once Steps 1 and 2 have converged at load step n, the newly collected training cases are temporarily added to the training database, and a cycle of training and update of neural network connection weights follows. This entire process, i.e. Steps 1 and 2, followed by a training cycle and update of neural network connection weights, is termed a local iteration. At each new local iteration, the training cases (σn−1+Δσn, εn−1+Δεn+δεn) that are collected replace in the training database those that were generated in previous local iterations at load step n. Local iterations are continued until the selected convergence criterion on δ(Un vector) (Eqtn. 3) is satisfied; that is, until Step 2 is not required.
Once this satisfactory agreement is achieved at load step n, the process is repeated for load step (n+1), and so on, until the full range of applied loads is covered. The term load pass is used to denote the application of this algorithm over the full range of applied loads. It is again noted that the stress strain training cases that are derived from this algorithm are not initially representative of the actual stress strain response of the material. For this reason, the complete training of the neural network material model may require several load passes. Eventually, with additional load passes, Step 1 of the autoprogressive algorithm produces fields of stress strain pairs that are more nearly consistent with the measured structural load vs. displacement responses, the structural displacement correction that is applied in Step 2 gradually becomes smaller and smaller, and convergence is declared.
As the autoprogressive algorithm proceeds from one load increment to the next, a set of stress strain training pairs is collected at each load step from the converged solution. At the conclusion of one load pass, therefore, the entire training database consists of a subset of the pre-training cases plus one set of training cases for each load step. Because the connection weights are updated periodically during the load pass, these different sets of training cases actually correspond, in effect, to different material models. Generally, the most recently collected sets most faithfully represent the actual material behavior. Therefore, preferably some strategy is adopted to either retire or correct the older training sets. One elective strategy that has been developed is to use a moving window to select only those training sets that were collected over a specified small number of prior load steps.
In addition, in more complex material modeling situations that involve multiaxial path-dependent behavior, where it is necessary to include stress path information in the neural network input, additional measures may be useful to ensure that the stress path information is consistent with the current material model.
The schematic of
In step 15 B of the exemplary autoprogressive training method of
The stress data from analysis of step 15 A and the strain data from analysis of step B are extracted to form stress-strain pairs that approximate the field soil constitutive response. The neural network soil model is trained using these stress-strain pairs. The analyses of step B and the subsequent neural network material model training are referred to as an autoprogressive training cycle. The analyses of step B are repeated using the updated neural network soil model and a new set of stress-strain data is extracted for re-training of the neural network soil model in additional autoprogressive training cycles.
The solution converges when the analysis of step 15 B. 1 provides the correct strain, and step 15 B.2 provides the correct stress. That is, when the modeled results converge at the measured values. Several autoprogressive training cycles are generally performed. This results in a single autoprogressive training pass. Several training passes are usually needed to develop a neural network material model that will adequately capture measured deformations in an FE analysis. Care should be exercised in choosing the number of autoprogressive training cycles and passes and the neural network training scheme. It is preferable that a small number of autoprogressive training cycles be used in the first few passes to allow the neural network material model to learn the full range of stress-strain response. This will prevent problems associated with early over training of the neural network material model.
It is noted that the procedure illustrated in
A preferred method of the invention also includes steps of using a nested adaptive neural network (NANN) architecture is used for representing soil constitutive behavior. NANN, with an example shown in
The input layer includes the strain vector and other relevant state variables (SV), and the output layer provides the stress vector. The two hidden layers are assigned an initial number of nodes, which can increase (i.e. adapt) to better fit the training data. For nonlinear material behavior, knowledge of loading history is needed to accurately capture material response. “History” modules can be added to include prior stress-strain states as additional input nodes. The following describes the governing equations of an exemplary NANN material model of the present invention.
The input layer consists of the updated strain vector εi and the current state variables SVi. The state variables can represent prior stress and strain states or other quantities deemed relevant to capture material behavior. In the first layer the strain vector may be scaled by an appropriate scaling factor:
The nodes of the first hidden layer are related to the nodes of the input layer using a set of connection weights:
Nε: Number of input strain nodes
wij Bε: connection weight between input node strain εj and 1st hidden layer node Bi
β: constant for the sigmoid function tanh
NSV: Number of state variable nodes
wij BSV: connection weight between input node state variable SVj and 1st hidden layer node Bi
State variables (SVj) that represent strain and stress quantities are scaled in accordance with Eqtn. 4 and Eqtn. 5, respectively.
The nodes of the second hidden layer of the exemplary NANN of the invention are similarly related to the nodes of the first hidden layer using another set of connection weights:
NB: Number of 1st hidden layer nodes B
wij CB: connection weight between 1st hidden layer node Bj and 2nd hidden layer node Ci
The nodes of the output layer, which represent the scaled stress vector, are related to the nodes of the second hidden layer using another set of connection weights:
NC: Number of 2nd hidden layer nodes C
wij CB: connection weight between 2nd hidden layer node Cj and output layer node stress
The stress vector is then calculated from the scaled stress vector using the following equation:
In methods and systems of the invention, NANN material models can be made to describe a specific material behavior by training the neural network using a set of data that contains material specific stress-strain pairs as well as related state variables. For a NANN material model each data set contains strains and related state variables that are fed to the neural network input nodes, and are then forwarded to the output, stress, nodes. These neural network stresses are compared with target stresses to calculate the error. Connection weights of the network are updated while the errors are back-propagated. Hence, multi-layer neural network is often called back-propagation neural network.
For a NANN model of the invention, the base module is first trained to capture approximate stress-strain relationships. Then, the first history module is added to consider the effect of loading history. Preferably, there are no connections from the base module to the first history module to eliminate the effect of later stress-strain states on previous stress-strain states (history). On the other hand, connections are added from the first history module to the base module, because later states are dependent upon the previous stress-strain states (history). When training the added first history module, the base module connection weights are fixed. More history modules can be added to capture highly nonlinear material behavior.
Many variations of the back-propagation processes have been proposed to update connection weights of the neural networks. The Rprop process as taught in “A Direct Adaptive Method for Faster Backpropagation Learning: The RPROP Algorithm,” Riedmiller, M. and H. Braun, IEEE International Conference on Neural Networks. (1993) San Francisco: IEEE, New York: pp.586-591, is an example of a back propagation process useful for training NANN models of the invention because of the algorithm's generally high reliability and relatively small number of control parameters.
Preferably, methods of modeling material behavior of the invention that use the neural network and NANN models also include a finite element (FE) analysis. A general schematic flow chart illustrating the relationship of an exemplary material constitutive model to the global solution of equilibrium equations in nonlinear finite element analysis is illustrated in
In incremental plasticity models the constitutive equations relate an infinitesimal strain increment, strain rate, (dε) with an infinitesimal stress increment, stress rate, (dσ) in the form of constitutive differential equations. The FE material module has to provide material response over a finite strain increment (Δε) and therefore the material constitutive equations have to be integrated. The basic assumption for integration of rate constitutive equations over the increment n+1Δε is that the material response is proportional over this strain increment:
Both implicit and explicit integration schemes have been used for numerical integration of elasto-plastic constitutive material models. For neural network based material models described by Eqtn. 1 through Eqtn. 8 and generally illustrated in
The material model in an FE analysis of the invention may provide the material stiffness matrix also known as the Jacobian. For infinitesimal strain increments (where Δε→dε), J is the continuum Jacobian, Jc:
For elasto-plastic constitutive models this matrix is the elasto-plastic material matrix and is explicitly defined in model constitutive equations. The numerical solution of the nonlinear FE equations may be obtained iteratively by Newton-Raphson method or a number of other implicit and explicit methods as will be appreciated by those knowledgeable in the art.
The neural network model provides a relationship of stress to strain and the implied material stiffness matrix can be extracted from this relationship. The material stiffness matrix (Jacobian) can be written as follows:
The above derivation demonstrates that a stiffness matrix can be derived for neural network material models of the invention. This stiffness matrix is dependent on the current state of the material and the strain increment. The matrix represents a consistent Jacobian obtained from the linearized material response over a desired strain increment. Eqtn. 11 represents an exact algorithmic linearization of the material stiffness matrix. The use of this matrix is believed to lead to efficient convergence of the Newton iteration in FE analysis useful for practice of the invention.
Methods and systems of the present invention contemplate that the neural network material model and proposed stiffness matrix can be used in a general FE analysis code using implicit methods such as the Newton-Raphson and similar methods. The proposed formulation and implementation is independent of the exact architecture of the neural network model and the training procedure.
The exemplary neural network constitutive model may be implemented in a general-purpose finite element code using the user defined material module or UMAT capability. Such an implementation makes it possible to use the exemplary material model in the solution of general boundary value problems. The implemented model is used in continuum stress analysis and rate independent static problems; therefore, the relevant convergence criteria of the global solution are related to nodal displacements and force residuals.
The derived material stiffness matrix for the neural network model shown in Eqtn 11 is independent of the specific material response learned by the neural network model. The stiffness matrix formulation can be used whether the neural network represents mildly or highly nonlinear material response. This is a distinct advantage over conventional elasto-plastic models whereby the material stiffness matrix has to be developed for each specific model.
The material stiffness matrix for an elasto-plastic constitutive model has characteristics directly related to the model formulation. The stiffness matrix is symmetric for a model with an associated flow rule and is asymmetric for a model with a non-associated flow rule. The flow rule is a mathematical idealization of measured or assumed material response. The mathematical formulation of a neural network material model does not explicitly account for symmetric or non-symmetric behavior. The neural network model learns the material behavior from the training data set provided. If the data set contains sufficient information about symmetric material behavior then the model will learn this behavior and the material stiffness will be symmetric, otherwise symmetry in the stiffness matrix is not guaranteed. It is believed that for an isotropic material the material behavior and consequently the stiffness matrix are symmetric. If the training data set does not contain sufficient information about material isotropic behavior, the training data set can be generalized to reflect the known material characteristic.
D. Field Observation Data Methods
Still an additional aspect of the present invention is directed to methods for using empirical three dimensional non-uniform field observation data to train a neural network based model to represent the field material. In practice, the neural network model discussed herein above, including its NANN and finite element aspects, will be appropriate for practice of field observation embodiments of the invention. Field observations may be made at construction sites where excavation, construction, or other activities are occurring, for instance, and may represent non-uniform strain, stress, force, and/or displacement data.
Measurements may be made over a period of time of days, weeks, months, or even years, depending on the application and scope of construction. By way of particular example, at a given excavation stage, the soil is excavated to a known depth and struts are placed to support the excavation wall. The measured deformations and the corresponding known excavation stage represent complementary sets of field observations. Similarly, as a building or other structure is built displacement of soil and other supports may occur. A steel beam supporting the base of the building, for example, may be displaced over time as the building is erected. Through methods of the invention, a self-organizing computational model of these and other problems will be developed to correctly represent the field observations. A preferred self-organizing computational model comprises a neural network consistent with that discussed herein with reference to other invention embodiments.
These field observation embodiments of the invention also contemplate use of an autoprogressive training method in combination with the preferred neural network model. The autoprogressive training method contemplated is consistent with that discussed herein above, except that the empirical training data includes field measurements or observations of actual material stress and strain instead of data obtained with a testing device.
It is intended that the specific embodiments and configurations herein disclosed are illustrative of the preferred and best modes for practicing the invention, and should not be interpreted as limitations on the scope of the invention as defined by the appended claims. For example, although discussion has been made herein of practice of the invention with soil, it will be appreciated that the invention will be beneficial when used with other materials, with examples including but not limited to granular materials such as sand, powders, grains, and the like.